Abstract: We derive the properties and demonstrate the desirability of a model-based method for estimating the spatially varying effects of covariates on a quantile function. By modeling the quantile function as a combination of I-spline basis functions and Pareto tail distributions, we allow for flexible parametric modeling of the extremes, while preserving the nonparametric flexibility in the center of the distribution. We further establish that the model guarantees the desired degree of differentiability in the density function, and enables us to estimate nonstationary covariance functions that are dependent on the predictors. We use a simulation study to show that the proposed method outperforms other methods in terms of producing efficient estimates of the effects of predictors, particularly in distributions with heavy tails. To illustrate the utility of the model, we apply it to measurements of benzene collected around an oil refinery to determine the effect of an emission source within the refinery on the distribution of the fence line measurements.

Key words and phrases: Conditional density estimation, quantile regression, spatially-varying coefficients.